Why are sine, cosine and tangent defined as opposite/hypotenuse ratios?

[iknownth]

We all know that
sin theta = opposite side / hypotenuse
cos theta = adjacent side / hypotenuse
tan theta = opposite side / adjacent side.
But why? Are there some explanations behind or are they just defined by scientists?

 

[pwsnafu]

But why? Are there some explanations behind or are they just defined by scientists?

What do you mean by "explanations"?

To answer the question, they are the definitions used in planar geometry. There are other (equivalent) ways of defining them, but you'll get the same properties nonetheless.

 

[iknownth]

How can one prove that sin theta = opposite side / hypotenuse ?

 

[pwsnafu]

How can one prove that sin theta = opposite side / hypotenuse ?

I'm assuming planar geometry here. There are two answers:

1. It is the definition of sine. There is nothing to prove.
2. The definition is using the unit circle. It which case the proof is immediate from similar triangles.

I guess it's worth asking: what is your definition of sine?

 

[micromass]

It is the definition of sine. There is nothing to prove.

In this case, there is still something to prove. You want to prove also that if two triangles have the same angle, then the quantity opposite side/hypothenuse is the same.

 

[pwsnafu]

In this case, there is still something to prove. You want to prove also that if two triangles have the same angle, then the quantity opposite side/hypothenuse is the same.

Arrgh yes of course.

Where's the brainfart smiley when you need one?